Problem 72 - Algebra 1

Question

Which of the following expressions is equivalent to the expression \(\frac{6 x-4}{-\frac{2}{3}} ?\) (A) \(2(3 x-2) \cdot \frac{3}{2} \quad\) (B) \((6 x-4) \cdot\left(-\frac{2}{3}\right) \quad\) (C) \(-9 x+6 \quad\) (D) \(9 x-6\)

Step-by-Step Solution

Answer

The expression is equivalent to option (C): -9x + 6.

Step 1: Simplify the given expression

First we need to simplify the given expression, \(\frac{6x-4}{-\frac{2}{3}}\). To do this, multiply both numerator and denominator by -3/2. Hence, the expression simplifies to \( (6x - 4) \cdot -\frac{3}{2} \).

Step 2: Distribute and simplify

Next, distribute -3/2 into (6x - 4). Here you will get -9x + 6.

Step 3: Compare with options

Now take the result from the previous step and compare it with the options given in the question. You can see that our result matches with option (C).

Key Concepts

Simplifying Expressions

Simplifying expressions is the process of turning complex algebraic equations into their simplest form. This is a key skill in algebra, allowing you to better understand and solve equations. Let's take the expression \( \frac{6x-4}{-\frac{2}{3}} \) as an example. It appears complex initially, due to the fraction.
To simplify, you multiply the numerator and the denominator by the reciprocal of the denominator. In this case, you multiply by \(-\frac{3}{2}\), turning \( \frac{6x-4}{-\frac{2}{3}} \) into \((6x - 4) \cdot -\frac{3}{2} \). Now, the expression is in a much more manageable format, making it easier to solve or compare with other expressions.

• Begin by identifying and reorganizing fractions or variables.
• Break down the problem by performing operations separately.
• Combine like terms where possible to further simplify.

Working to simplify algebraic expressions is crucial for making them more understandable and getting them ready for additional steps.

Distributive Property

The distributive property is a foundational concept in algebra that involves distributing a multiplier across terms within parentheses. This rule lets you rewrite expressions in a different but equivalent form. In this exercise, you would take the expression \((6x - 4) \cdot -\frac{3}{2} \) and apply the distributive property.
To apply it, multiply \(-\frac{3}{2} \) by each term inside the parentheses separately:

• Multiply \(-\frac{3}{2} \) by \(6x\): This results in \(-9x\).
• Multiply \(-\frac{3}{2} \) by \(-4\): This gives \(+6\).

Putting these results together, you get the expression \(-9x + 6\).
Applying the distributive property helps break down and solve more complicated problems by allowing a step-by-step approach. This is particularly useful when dealing with multiple terms within brackets, ensuring that each part of the expression is multiplied accurately.

Numerator and Denominator

Understanding numerators and denominators is essential when working with fractions in algebraic expressions. The numerator is the top part of a fraction, representing the number of parts considered out of a whole indicated by the denominator.
The original problem involves a complex fraction: \( \frac{6x-4}{-\frac{2}{3}} \).

• The "6x - 4" is the numerator, a term that you work with directly in expressions.
• The "-\frac{2}{3}" serves as the denominator, which tells how to divide the numerator or understand its fraction form more accurately.

In algebra, manipulating the numerator and denominator properly is essential. For instance, simplifying \( \frac{6x-4}{-\frac{2}{3}} \) involves multiplying by the reciprocal of the denominator to transform the fraction into a simpler, numerator-led equation.
Understanding and working with numerators and denominators ensure you can accurately compare, simplify, or even solve mathematical expressions involving fractions.